This paper proposes an approach to solve the transient stability constraint optimal power flow (TSC-OPF) problem. The transient stability constraints are expressed as the critical clearing time (CCT) of different contingencies, and are approximated using artificial neural networks (ANNs). The ANNs provide a nonlinear, differentiable mapping between the load flow variables and the CCT. As a result, the TSC-OPF with multiple transient stability constraints can be solved very efficiently with little additional computational burden. The effectiveness of the proposed method is demonstrated with the IEEE 39 bus and the IEEE 300 bus systems.